Question

Let \(\mathrm{P}(\mathrm{r})\left[\mathrm{Q} /\left(\pi \mathrm{R}^{4}\right)\right] \mathrm{r}\) be the charge density distribution for a solid sphere of radius \(\mathrm{R}\) and total charge \(\mathrm{Q}\). For a point ' \(\mathrm{P}\) ' inside the sphere at distance \(\mathrm{r}_{1}\) from the centre of the sphere the magnitude of electric field is (A) \(\left[\mathrm{Q} /\left(4 \pi \epsilon_{0} \mathrm{r}_{1}^{2}\right)\right]\) (B) \(\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(4 \pi \in{ }_{0} \mathrm{R}^{4}\right)\right]\) (C) \(\left[\left(\mathrm{Qr}_{1}^{2}\right) /\left(3 \pi \epsilon_{0} \mathrm{R}^{4}\right)\right]\)

Short answer

The short answer is: (B) \[\frac{Qr_{1}^2}{4\pi \epsilon_{0} R^4}\]

Step-by-step solution

Gaussian Surface

To apply Gauss's law for a solid sphere with charge density distribution, we need to choose an appropriate Gaussian surface. In this case, we will choose a sphere with radius r₁ centered at the center of the charged sphere because of the symmetry.

Setup Gauss's Law

Gauss's law states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space (ε₀). Mathematically, it's given by: \[\oint \vec{E} \cdot d\vec{A} = \frac{Q_\text{enclosed}}{\epsilon_{0}}\] Where \(\vec{E}\) is the electric field vector, \(d\vec{A}\) is the area vector of the Gaussian surface, and \(Q_\text{enclosed}\) is the total charge enclosed by the Gaussian surface.

Calculate Enclosed Charge

To find the enclosed charge, we need to integrate the charge density over the volume enclosed by our Gaussian surface (a sphere with radius r₁): \[Q_\text{enclosed} = \int_{V} P(r) dV\] Where \(P(r) = [\frac{Q}{\pi R^4}]r\) and \(dV = 4\pi r^2 dr\). Integrate over the volume of the Gaussian sphere (from 0 to r₁): \[Q_\text{enclosed} = \int_{0}^{r_{1}} \left[\frac{Q}{\pi R^4}\right]r(4\pi r^2 dr)\]

Solve Integral

Performing the integration, we get: \[Q_\text{enclosed} = \frac{4Q}{R^4}\int_{0}^{r_{1}} r^3 dr = \frac{4Q}{R^4}\left[\frac{r_{1}^4}{4}\right] = \frac{Qr_{1}^4}{R^4}\]

Apply Gauss's Law

Now, we can apply Gauss's law for our Gaussian sphere: \[\oint \vec{E} \cdot d\vec{A} = \frac{Q_\text{enclosed}}{\epsilon_{0}}\] Due to the spherical symmetry, the electric field has the same magnitude at every point on the Gaussian surface and is parallel to the area vector. Thus, their dot product simplifies to their magnitudes multiplied together: \[\oint E dA = E\oint dA = E \cdot 4\pi r_{1}^2 = \frac{Q_\text{enclosed}}{\epsilon_{0}}\] Let's substitute the value of \(Q_\text{enclosed}\) from Step 4: \[E \cdot 4\pi r_{1}^2 = \frac{Qr_{1}^4}{R^4\epsilon_{0}}\]

Solve for Electric Field Magnitude

Finally, we solve for the electric field magnitude E: \[E = \frac{Qr_{1}^2}{4\pi R^4\epsilon_{0}}\] Now let's compare this result with the given options. We can see that our result matches option (B), so the correct answer is: (B) \[\frac{Qr_{1}^2}{4\pi \epsilon_{0} R^4}\]

Key concepts

Electric Field

In physics, the electric field is a fundamental concept that describes the influence that a charged object exerts on other charged objects around it. It is represented by the symbol \( \vec{E} \) and is a vector quantity. This means it has both magnitude and direction.

The electric field at a point is defined as the force experienced by a unit positive charge placed at that point, divided by the magnitude of the charge. Mathematically, this can be expressed as:
\[ \vec{E} = \frac{\vec{F}}{q} \]
where \( \vec{F} \) is the force in newtons, and \( q \) is the charge in coulombs.

• *Magnitude*: Measured in newtons per coulomb (N/C).
• *Direction*: Defined as the direction a positive charge would move.

The electric field is a crucial part of understanding electric forces and interactions and is key in applying Gauss's Law to determine field distributions.

Charge Density

Charge density describes how electric charge is distributed in a given system. It is defined as charge per unit volume, area, or length, depending on the distribution. In this context, we are concerned with volume charge density, denoted by \( \rho \).

The exercise uses a specific charge density distribution for a sphere, given by:
\[ P(r) = \left[ \frac{Q}{\pi R^4} \right] r \]
where \( Q \) is the total charge, \( R \) is the radius of the sphere, and \( r \) is the distance from the center.

• *Uniform vs. Non-uniform*: Our distribution is non-uniform as it varies with \( r \).
• *Application*: Understanding the charge density helps in calculating the enclosed charge for Gauss's Law.

By integrating this charge density over a volume, we find the total charge within a specific region, which is essential for calculating the electric field.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in Gauss's Law to simplify the calculation of electric fields. The choice of the Gaussian surface is crucial, as it should ideally allow you to take advantage of the symmetry in the problem.

In this exercise, we use a spherical Gaussian surface of radius \( r_1 \) centered at the sphere's center.

• *Symmetry:* Spherical symmetry simplifies calculations, as the electric field is the same at every point on the surface.
• *Purpose:* Helps apply Gauss's Law to calculate the electric field by finding the total flux through the surface.

Gauss's Law relates the electric flux through this surface to the enclosed charge, allowing us to find electric field magnitudes efficiently.

Permittivity of Free Space

The permittivity of free space, denoted as \( \epsilon_0 \), is a physical constant that describes how electric fields interact in a vacuum. It plays a critical role in the formulation of various equations in electromagnetism, including Gauss’s Law.

The permittivity of free space has a defined value of \( 8.854 \times 10^{-12} \) farads per meter (F/m).

• *Role in Gauss's Law:* Appears in the denominator of the formula, influencing the calculation of electric fields.
• *Unit:* Its unit is farads per meter, reflecting its function in determining how much electric field is permitted in a vacuum for a given amount of charge.

This constant is fundamental in the equations used to calculate electrical interactions and field strengths, particularly when evaluating how fields permeate through vacuum or space.